Plane partitions (III): The weak Macdonald conjecture

Abstract

Thus the plane partitions of 3 are 3, 21, 2, 111, 11, 1 1 1 1 1. There are some well-known theorems and open questions related to plane partitions in which the number of rows and columns and the size of the parts is restricted. I.G. Macdonald [10] has devised a notation that allows a uniform consideration of these questions. First one considers the "Ferrers graph" D(zr) of a plane partition ~; this is the set of integer points (i,j, k) in the first octant that satisfy l < k < a i j . Next define the height of p=(i,j, k) to be ht(p)=i+j+k-2. Defining ~l . . . . =[1, 1] x [1, m] x [1, n] ~TZ 3, (1.1)